Is Partial Fractions Necessary?

$\large\mathfrak{S}= \sum_{n=1}^\infty\left(\dfrac{2n+1}{n^2(n+1)^2}\right)$

$\large {\begin{aligned}\mathfrak S&=\sum_{n=1}^\infty\left(\dfrac{(n+1)^2-n^2}{n^2(n+1)^2}\right) \\&=\sum_{n=1}^\infty\left(\dfrac{1}{n^2}- \dfrac{1}{(n+1)^2}\right) \end{aligned}}$

$\large(\mathbf{A~Telescopic~Series})$

$=\dfrac{1}{1^2}-\cancel{\dfrac{1}{2^2}}+\cancel{\dfrac{1}{2^2}}-\cancel{\dfrac{1}{3^2}}+\cancel{\dfrac{1}{3^2}}\cdots$

$\huge =\dfrac{1}{1^2}=\boxed 1$